Question: A spring is hanging from a ceiling. The length $L(t)$ (in $\text{cm}$ ) of the spring as a function of time $t$ (in seconds) can be modeled by a sinusoidal expression of the form $a\cdot\sin(b\cdot t)+d$. At $t=0$, when the spring is exactly in the middle of its oscillation, its length is $7\text{ cm}$. After $0.5$ seconds the spring reaches its maximum length, which is $12\text{ cm}$. Find $L(t)$. $\textit{t}$ should be in radians. $L(t) = $
Answer: The strategy First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph. Then, we should use the given information to find the amplitude, midline, and period of the function's graph. Finally, we should find $a$, $b$, and $d$ in the expression $a\sin(b\cdot t)+d$ by considering the features we found. Converting the given information into mathematical terms At $t=0$, the spring is $7\text{ cm}$ long. This means the graph of the function passes through $(0,7)$. We are given that this is the middle of the swing, which corresponds to the midline of the graph. $0.5$ seconds later (which means $t=0.5$ ) the length is $12\text{ cm}$. This corresponds to the point $(0.5,12)$. We are given that this is the maximum length, which corresponds to a maximum point of the graph. In conclusion, the graph intersects its midline at $(0,7)$ and then has a maximum point at $(0.5,12)$. Determining the amplitude, midline, and period The midline intersection is at $y={7}$, so this is the midline. The maximum point is $5$ units above the midline, so the amplitude is ${5}$. The maximum point is $0.5$ units to the right of the midline intersection, so the period is $4\cdot 0.5={2}$. [Why did we multiply by 4?] Determining the parameters in $a\sin(b\cdot t)+d$ Since the midline intersection at $t=0$ is followed by a maximum point, we know that $a>0$. [How do we know that?] The amplitude is ${5}$, so $|a|={5}$. Since $a>0$, we can conclude that $a=5$. The midline is $y={7}$, so $d=7$. The period is ${2}$, so $b=\dfrac{2\pi}{{2}}=\pi$. The answer $L(t)=5\sin\left(\pi t\right)+7$